Nowadays, many sisters carry bags or have phone cases or similar items with Quran verses written on them, or with ‘La ilaha illa Allah’ or similar phrases, or images of the Kaaba. Is this permissible?

No. What if they go to the bathroom? What if it falls on the ground and somebody steps on it? What if children play with these things? There are so many "what ifs." The Quran and the name of Allah (Subhanahu wa ta'ala) should not be subjected to insult (imtihan) by being taken into dirty places, falling on the ground where people can step on them, or being used merely for decoration. The words of Allah (Subhanahu wa ta'ala), the Quran, and the Sunnah are there as guidance for mankind. We must learn them, understand them, translate them into action, and make them our whole lives, rather than putting them up as decorations, gadgets, and so on.